17

CHAPTER 2: WAVEGUIDE ANALYSIS METHODS

2.1 Introduction

This chapter discusses the types of the waveguides and the behavior of light from the

point of view of ray optics and electromagnetic theory, followed by a brief treatment on

simulation techniques. Subsequently, the basic idea of the Finite Difference Method

(FDM) will be elaborated further in Section 2.2 as will the effects of the boundary

conditions. This is followed by a discussion on Beam Propagation Method (BPM) and

how FDM is employed in BPM. The last section describes the Coupled Mode Theory

(CMT) which is used to deal with coupling activities between two closely spaced

waveguides.

2.2 Planar Waveguides

Planar waveguides are optical components that allow the confinement of light within

certain boundaries by total internal reflection. Typically, there are three main types of

basic waveguides structure commonly used in silica-on-silicon platform: ridge, rib, and

buried waveguide as illustrated in Figure 2.1 [1]. In Figure 2.1, the waveguide core

layer is shown in dark blue whereas the waveguide cladding is shaded in light blue.

18

Figure 2.1: Three planar waveguide types: (a) ridge waveguide; (b) rib waveguide;

and (c) buried waveguide.

The core layer has a higher refractive index than its surrounding cladding layer in order

to keep the light (in the form of photon) well confined in the waveguide. It thus can be

said that the cladding is treated as a protective layer to guide the light within the core

with minimum losses and throughout this work, the buried waveguide is used. The

mechanism of light guidance in the waveguide will be discussed in the next section.

2.2.1 Light Behavior from the Point of View of Ray Optics

Light behavior within a planar waveguide can be outlined and analyzed

from the point of view of ray optics. Consider the planar waveguide illustrated in

Figure 2.1(c), where core layer (dark blue) is enclosed or sandwiched by

cladding layer (light blue). We assume n1 to represent the refractive index value

of the core layer and n2 corresponding to the refractive index value of the

cladding layer. In addition, we also assume that n1 is higher than n2. Thus, the

propagation of light through the planar waveguide by Total Internal Reflection

(TIR) can be intuitively understood by the use of the ray optics model.

Light source

(a) (b) (c)

Core Cladding

19

As shown earlier, the flow of light within the waveguide is governed by Snell’s

Law. Snell’s Law is used to describe the relationship that governs the light

propagation, where incident angle 1 of light in a medium with index n1 impinge

on the boundary of a dissimilar medium with index n2, resulting the light is

refracted at angle of 2 as depicted in Figure 2.2 (a).

The relationship that governs the light propagation that was derived from Snell’s

Law is shown below [2];

2211 sinsin nn (2.1)

is measured with respect to the normal of interface of two different medium.

In order to confine light within the waveguide, n1 has to be larger than n2.

Furthermore, if 2 becomes 90o, a condition termed total internal reflection (TIR)

occurs, where incident light impinging on the boundary is reflected back into

same medium. The incident angle, 1 that allow for this situation is called the

critical angle, c . As such, equation 2.1 can be simplified to [2];

n1

n2

1

2 If 1 > c

cladding, n2

cladding, n2

core, n1

c

n n1

n2

n2

(a) (b) (c)

Figure 2.2: (a) Reflection and refraction at a plane interface, (b) propagation of light

through optical waveguide by total internal reflection (TIR), (c) refractive index

profile of the optical waveguide

x

20

)(sin1

21

1n

n (2.2)

It can be seen from Fig. 2.2(b), if incident angle 1 exceed c , light will continue

to be reflected at the interface and subsequently guided via TIR in the core layer

[2]. The distribution of the refractive index which core layer has higher index

value is illustrated in Fig. 2.2(c). It is found that a small portion of guided light is

accumulated outside of core region. It is called evanescent field and this is very

useful for energy transfer to another adjacent waveguide.

2.2.2 Light Behavior from the Point of View of Electromagnetic Theory

Although the ray optics approach can be utilized for qualitative description and

basic understanding for light behavior within the waveguide, the technique lacks

the information and ability to explain the relationship between the electric and

magnetic field distribution. In many applications such as optical coupling, it is

essential to know the distribution of optical field or intensity within the

waveguide. Therefore it is necessary to include the more rigorous waveguide

treatment based on Maxwell’s equations.

The electromagnetic theory of light applied to the planar waveguide is

described by Maxwell’s equations [3-5]. We start the Maxwell’s equations by

assuming that the light is penetrating via a non-conductive dielectric

(conductivity 0 ), isotropic, non-magnetic (magnetic permeability o ),

and linear medium ( ED ). Therefore, Maxwell’s equations are reduced to [5]:

21

t

HEX o

(2.3)

t

EnHX o

2 (2.4)

where E

and H

are the electric and magnetic fields respectively, o is the free

space permeability, o is the permittivity of the free space and n is the refractive

index of the light propagation medium. Under the conditions mentioned above, it

is noted that div E and div H are equal to zero.

For an optically inhomogeneous medium with the refractive index

changes only in the transverse direction, )(rnn and using Maxwell’s

equations (2.3) and (2.4), we obtain the following wave equations for E

and H

[5]:

0)1

(2

222

2

2

t

EnEn

nE oo

(2.5)

0)(1

2

222

2

2

t

HnHXXn

nH oo

(2.6)

The above equations indicate that the Cartesian components of the

electric field vector xE , yE and zE and magnetic field vector xH , yH and zH are

coupled in an inhomogeneous medium. In this regard, a scalar wave equation for

each component cannot be established as in the case of a homogeneous medium.

However, the adequate solution to the inhomogeneous wave equations (2.5) and

(2.6) for monochromatic wave is described by the form [5]:

22

)(),(),( ztieyxEtrE

(2.7)

)(),(),( ztieyxHtrH

(2.8)

where is the propagation constant and being the angular frequency of the

wave. These two expressions indicate the electromagnetic field for a propagating

mode. By applying the solution for planar waveguide given in equations (2.7)

and (2.8) to Maxwell’s equations in equations (2.3) and (2.4), and taking their x,

y and z components as shown in Figure 2.3, we obtain the following expressions

[5]:

xoyz HjEj

y

E

(2.9)

zoxy

Hjy

E

x

E

(2.10)

yoz

x Enjx

HHj 2

(2.11)

xoyz EnjHj

y

H 2

(2.12)

zoxy

Enjy

H

x

H2

(2.13)

yoz

x Hjx

EEj

(2.14)

23

In the case of wave guiding confined to one direction (one dimension

confinement), the confinement is assumed to be in x-direction within the core

layer. In addition, the upper and lower cladding layers are respectively assumed

to be infinity. Furthermore, the light is assumed to propagate in z-direction and

the waveguide is also assumed no dependence in y-direction. Therefore the

electric and magnetic fields in the planar waveguide show independence on y-

direction and by setting 0

y

E

and 0

y

H

equations (2.9) to (2.14) can be

simplified to [5]:

xoy HE (2.15)

zo

yHj

x

E

(2.16)

yoz

x Exnjx

HHj )(2

(2.17)

xoy ExnH )(2 (2.18)

zo

yExnj

x

H)(2

(2.19)

yoz

x Hjx

EEj

(2.20)

x z

y

Core, n1

Upper cladding, n3

Lower cladding, n2

0

-d

Figure 2.3: The basic structure of the slab waveguide where the core layer is

sandwiched between upper cladding and lower cladding

24

The first three equations only involve yE , xH and zH and the last three

expressions involve xE , zE and yH . The first set of equations, equations (2.15)

to (2.17) corresponds to the TE (transverse electric) modes as TE modes only

contain the transverse component ( yE ) with respect to the direction of

propagation z, thus 0zE . Meanwhile the second set of equations are denoted as

TM (transverse magnetic) modes which having the non-vanishing value of xE ,

zE and yH . Similarly, the magnetic fields have only a transverse component

( yH ) and no magnetic field along the z direction, hence 0zH .

As discussed earlier, the propagation of light within a planar waveguide

may be described in terms of TE modes and TM modes. We first consider the

TE modes by substituting xH and zH components from equations (2.15) and

(2.16) into equation (2.17) to obtain the TE wave equation for yE as [5]

0])([ 222

2

2

yo

yExnk

dx

E (2.21)

where the free space wave number is given by, ooook .

The analysis so far is valid for an arbitrary x-dependent profile. Referring

back to Figure 2.3, the core layer which has a refractive index value, 1n is

sandwiched between two cladding layers which have refractive index values

2n and 3n . They are separated by planar boundaries perpendicular to the x-axis.

In this case, z is the propagation axis for the light beam. In order to confine the

light beam to within the waveguide, we assume 321 nnn . The plane

25

0x corresponds to the upper cladding-core boundary and the core-lower

cladding boundary is located at dx . Thus the core thickness is d . The

specific index profile of planar waveguide structures as depicted in Figure. 2.3 is

utilized and given as below:

;

;

;

)(

3

2

1

n

n

n

xn

0

0

x

dx

xd

(2.22)

The mode that is mainly confined within the core layer is denoted as guided

modes and its power decay exponentially in the cladding. To fulfill boundary

condition at 0 xd , the propagation constant associated with a particular

mode must have:

12 nknk oo (2.23)

In term of the refractive index, the effective index N of guided mode must be

located between 1n and 2n :

12 nNn (2.24)

where ok

N

. By imposing suitable boundary conditions, the wave equation

from (2.21) can be written as [5]:

26

02

12

2

y

yE

dx

E 0x (upper cladding) (2.25)

02

22

2

y

yE

dx

E dx (lower cladding) (2.26)

02

2

2

y

yE

dx

E 0 xd (cover) (2.27)

where the two parameters and are shown follows:

2

3

222

1 nko (2.28)

22

1

22 nko (2.29)

2

2

222

2 nko (2.30)

The solution of the electric field in the upper cladding, core and lower cladding

from equation (2.25) - (2.27) can be presented as [5]:

yEx

xixi

x

De

CeBe

Ae

2

1

dx

xd

x

0

0

(2.31)

where A , B , C and D are constant. Applying the continuity of yE and dx

dE yat

the boundaries 0x and dx yields the four equations that related to

constants A , B , C and D and propagation constant [5]:

ACB (2.32)

27

dididi DeCeBe 1 (2.33)

ACiBi 1 (2.34)

ddidi DeCeiBei 2

2

(2.35)

By solving equations (2.32) to (2.35), the following equation is acquired [5]:

21

21

1

tan d

(2.36)

The relation above can be considered as the dispersion relation for the

asymmetric step index planar waveguide and is a transcendental equation which

will determine the values of propagation constant. For the case of symmetric

step index planar waveguide ( 32 nn ), the transcendental equation is reduced to:

2

1

1

1

2

tan

d (2.37)

where 21 . The following descriptions only take account of asymmetric step

index planar waveguide. It is simple to modify the asymmetric expressions to

symmetric expression. In order to transcendental equation in equation (2.36) can

be universalized for any asymmetric step index waveguide, it is convenient to

introduce a set of normalized parameters:

28

2

2

2

1

2

2

2 )(

nn

nNb

; Normalized mode index, (2.38)

2

1

2

2

2

1 )( nndkV o ; Normalized core thickness/V-number, (2.39)

and 2

2

2

1

2

3

2

2 )(

nn

nna

; Asymmetry measure. (2.40)

The effective index N corresponding to a confined mode is in the range of

12 nNn whereas the normalized mode index b is bounded between 0 and 1.

As deduced from equation (2.39), the normalized core thickness V or V-number

is proportional to the thickness of core layer d and inversely proportional to

wavelength . The V-number includes all the waveguide parameters that will

determine the guidance behavior of the waveguide. On the other hand, the

asymmetry measured at a is zero in the case of symmetric waveguide,

thus 32 nn .

By rearranging the transcendental equation, it can be rewritten as the

function of propagation constant and in terms of the normalized parameters

[5]:

)1(

)(1

111tan

b

abb

b

ab

b

b

bV

(2.41)

The electric field in the three different regions can be determined after the

propagation constant is obtained numerically from equation (2.41):

29

yE

)(1

1

2

1

)sin(cos

)sin(cos

dx

x

eddA

xxA

Ae

dx

xd

x

0

0

(2.42)

Based on expressions above, the electrical fields decrease exponentially in upper

and lower cladding layers whilst the electrical fields vary sinusoidal in the core

layer. The solution for yE is completely validated except the constant A . It is

related to the energy carried by the mode.

Following the same procedure in determining yE , the magnetic field

component yH of a particular guided mode can be obtained. The wave equation

for TM propagation mode is identical to that obtained in TE, with the exception

that the magnetic field function has been established. TM wave equation for

yH is given as below:

0])([ 222

2

2

yo

yHxnk

dx

H (2.43)

The transcendental equation for TM waveguide is obtained in terms of

normalized parameters [5]:

)1(

)(11

1

1

1

1

1tan

b

abb

b

ab

b

b

bV

ba

ba

(2.44)

30

For the sake of simplicity, the a and b are respectively defined as

2

1

2

n

na and )1(

2

1

3bab a

n

n

. The solutions for the magnetic field

are [ref]:

yE

)(1

2

3

2

1

1

2

3

2

1

2

1

)sin(cos

)sin(cos

dx

x

edn

ndA

xn

nxA

Ae

dx

xd

x

0

0

(2.45)

The objective of the above discussion is to provide both qualitative and

quantitative picture about the guidance behavior of the waveguide. Nevertheless,

the theoretical treatment above is only restricted to the slab waveguide structure

as depicted in Figure 2.3. In the latter section, we will discuss the methods which

can be implemented for the guidance in channel waveguide. There are numerous

methods are available including effective index method [6, 7], Marcatili’s

method [8] and the FDM [9]. The first two techniques are considered as

analytical method and they will not be utilized throughout the work. It is found

that the analytical method is accurate than the numerical method. However the

FDM as a numerical method is employ in this work and will be elucidated at

next section.

31

2.3 The Finite Difference Method (FDM)

As discussed earlier, the electromagnetic fields propagating along the waveguide

are composed of guided modes, also can be addressed as eigenmodes, which are

dominated by transverse electric and magnetic field components with their

corresponding propagation constants. The behavior of these electromagnetic fields can

be analyzed and derived from the Maxwell’s equations. The analytical solution is

acquired by solving the Maxwell’s equations. Nevertheless, this is only for simple

waveguide geometries such as slab waveguides. For this reason, solving

electromagnetic problems for practical waveguide geometries requires the use of

numerical methods. Numerical methods are based on the approximation to the exact

solution and the standard that minimizes the error between the two. In the literature,

there are several numerical methods available such as FDM [10], Finite Element

Method (FEM) [11] and Method of Lines (MoL) [12] that can be used to solve the

eigenmode. Throughout this work, the FDM is employed to analyze the eigenmode

characteristics of an optical waveguide owing to its good numerical efficiency and

accuracy.

In the FDM, the cross-section of the waveguide is made discrete with a

rectangular grid of points which might be of identical or variable spacing as illustrated

in Figure 2.4. In each of the subdivisions, a two-dimensional wave equation is replaced

with appropriate Finite Difference relationship which is derived from a five-point

Taylor series formula [10]. Each grid of point is assigned to an arbitrary electric field

value. Due to the subdivisions being rectangular, thus the FDM is appropriate for

rectangular waveguide structure. As shown in Figure 2.4, by defining to be electric

field component to be calculated, the relationships are shown as below:

32

22

2 ),1(),(2),1(

x

nmnmnm

dx

, and (2.46)

22

2 )1,(),(2)1,(

y

nmnmnm

dy

(2.47)

where 2x and 2y are spacing between two grid of points in x and y direction

respectively.

There is a variety of boundary conditions that can be imposed at the edge of the analysis

window such as Dirichlet, Neumann [13] and Transparent Boundary Condition (TBC)

[14, 15]. The first and second boundary conditions for the calculation window are

categorized as fixed boundary condition. This means that the field (electric or magnetic

fields) is required to be set to zero at the boundary of the analysis window. It is a good

approximation if there is a large index discontinuity at the edge. Nevertheless, it

effectively reflects back the radiation to the analysis domain. To eliminate the back

reflections or incoming fluxes into the analysis window, the TBC is applied. It

effectively allows radiation to pass through the boundary freely and leave the analysis

domain without appreciable reflection. In this way, the unwanted interference in the

solution region (core layer) can be prevented [15].

x

n n+1 n-1

m

m+1

m-1

y

Figure 2.4: The cross-section of the waveguide is made discrete with a rectangular

grid of points which have identical spacing.

33

2.4 Finite Difference Beam Propagation Method

As discussed in the previous section, FDM shows an excellent way to solve the

waveguide eigenmode. Nevertheless it could not be utilized in solving the propagation

characteristic in integrated optics and fiber optics. Figure 2.5 shows the 3-Dimensional

Finite Difference as a plane rather than a ling along the z-axis.

The BPM is a widely used and indispensable numerical technique in today’s modeling

and simulation of evolution of electromagnetic fields in arbitrary inhomogeneous

medium. BPM is eligible to apply in complex geometries and automatically consider

both guided and radiation modes [10]. There are several numerical methods that can be

employed in BPM including Fast Fourier Transform Method (FFT) [16], (FEM [17] and

FDM [18]. In this project, the Finite Difference Beam Propagation Method (FD-BPM)

is employed to investigate light propagation in the silica based pump/signal MUX.

Z

Z+Z

Figure 2.5: The 3D FD algorithm propagates a plane rather than a line along the z-

propagation direction.

34

2.4.1 Beam Propagation Method

Simulation and design works involve approximation [19]. In short, the BPM

employs the FDM to solve the well-known parabolic or paraxial approximation

of Helmholtz equation [20]. BPM was proposed by Fleck et al. (1976) [21] for

solving the scalar Helmholtz equation. However BPM only started to be used in

analyzing and designing integrated optic devices in 1983 by Feit et al. [22].

Although BPM is a solution for paraxial forward propagating wave but it can be

expanded to include effects such as wide angle propagation via Padé

approximation, polarization effect and bi-direction propagation.

As discussed earlier, BPM is a particular approach for approximating the

exact wave equation for monochromatic waves and it can be solved numerically

by FDM [23]. In this section, the main features of BPM and its boundary

condition will be summarized by formulating the problem under restriction of

scalar field (neglect polarization effect) and paraxiality (propagation is restricted

to a narrow ranges of angles) [20]. The three-dimensional scalar wave equation

can be written in the form of Helmholtz equation for monochromatic wave as [4]:

0),,(22

2

2

2

2

2

2

Ezyxnkdz

E

dy

E

dx

E (2.48)

The slowly varying envelope approximation is used to approximate the electric

field ),,( zyxE in the +z direction. In this approximation, ),,( zyxE is separated

into two parts: the axially slowly varying envelope term of ),,( zyx and the

rapidly varying term of )exp( zjkno . Then ),,( zyxE can be expressed as:

35

)exp(),,(),,( zjknzyxzyxE o (2.49)

where the notation of on is a refractive index in the cladding. Substituting the

equation (2.49) into (2.48), we get

0)(2 2222

oo nnk

zknj (2.50)

where 2 is Laplacian equation and is expressed as

2

2

2

2

2

22

zyx

(2.51)

By assuming the weakly guiding condition ( )(2)( 22

ooo nnnnn ), the

equation (2.50) can be rewritten as:

)(2

1 2

o

o

nnjkkn

jz

(2.52)

The above is the wide-angle BPM equation. However, when 02

2

z

, the

equation (2.52) is reduced to:

)()(2

12

2

2

2

o

o

nnjkyxkn

jz

(2.53)

36

This is the para-axial 3D BPM equation. 2D BPM equation is obtained by

omitting the y-direction dependency.

2.4.2 Beam Propagation Method Based On Finite Difference

For simplicity, BPM analysis based on finite difference in this section will be

developed in 2D scalar Helmholtz scalar wave equation which is expressed as:

]),([2

),(2

1 22

2

2

o

oo

nzxnn

kjzx

xknj

z

(2.54)

In equation (2.54), )( 22

onn is not approximated as )(2 oo nnn [4]. Hence,

equation (2.54) can be used in both weakly guiding and strong guiding

conditions. In general, a differential equation of the form:

),(),(2

2

zxBx

zxAz

(2.55)

which can then be can be approximated by FDM as:

zz

m

i

m

i

1

, (2.56)

2

1

1

11

1

2

112/1

2

2

)(

2

)(

2

2

1),(

xxA

xzxA

m

i

m

i

m

i

m

i

m

i

m

im

i

, and (2.57)

37

)(2

1),( 12/1 m

i

m

i

m

iBzxB (2.58)

where x and z imply the compute step in the x- and z-axis respectively.

Meanwhile subscript i and superscript m are the respective grid point along the

x- and z-axis. Comparing equation (2.54) and (2.55), we get:

oknjA

2

1 (2.59)

]),([2

),( 22

o

o

nzxnn

kjzxB (2.60)

Substituting equation (2.56)-(2.60) into equation (2.54), we acquire the

following equations:

m

i

m

i

m

i

m

i

m

i

m

i

m

i

m

i

m

i dqs

11

1

1

11

1 , (2.61)

2/122

222/122 )(2)(4

])[()(2

m

ioo

o

m

i

m

i xknjz

xknjnnxks , and (2.62)

2/122

222/122 )(2)(4

])[()(2

m

ioo

o

m

i

m

i xknjz

xknjnnxkq (2.63)

Therefore when the initial electric field 0m

i is given at z=0, the electric field

m

i at the next propagation step, z=zm can be obtained by computing the equation

(2.61). The FD-BPM based on 3D Helmholtz wave equation can be found in the

following literature [4]. The Transparent Boundary Condition (TBC) [14] is

imposed to FDBPM instead of Dirichlet condition ( 0 ) and Neumann

38

condition ( 0

x

) to reduce reflection from incoming radiation field. TBC

serve as a boundary when the radiation reach the edge of simulation window, it

disappears into the boundary without any reflection.

2.5 Physics of Coupled Waveguide Device

An analysis for the coupling is a critical part to the design of a wide range of

coupled waveguide devices including directional coupler, Mach-Zender interferometer,

bragg grating, ring resonator and modulator. Hence, coupled mode theory is employed

to calculate the coupling along the interaction length and outputs at various wavelengths.

2.5.1 Coupling of Light between Waveguide

The coupling activities are taking place between two adjacent waveguides owing

to the waveguide mode evanescent fields overlapping. When the lights propagate

in a waveguide, small portions of mode or evanescent field propagate into

cladding region as illustrated in Figure 2.6.

(a) (b)

Light

source

Light

source

Figure 2.6: (a) the basic structure of two adjacent waveguide (b) when the lights

launch into an input and propagate down a waveguide, small portions of

evanescent field propagate into cladding region.

39

The intensity of the evanescent field decay exponentially with escalating the

distance into the cladding region [24]. The coupling activities occur between

waveguides only if both waveguides are placed sufficiently close to each other.

The coupling activities are due to the significant overlapping of evanescent field

between one waveguide to the adjacent one [25]. In this condition, the light

energy will be transferred completely from one waveguide to the other in a

periodic manner along the transmission direction [26]. The desired fraction of

light energy can be obtained at a specific length.

2.5.2 Coupled Mode Theory

Generally, the CMT is a method that can be utilized for dealing with the mutual

lightwave interaction between two propagation modes [4]. The behavior of two

modes having mutual coupling is described by the Maxwell’s equations.

Nevertheless a basic understanding of mutual coupling can be acquired from

coupling mode equations [4]:

zi abezBidz

zdA )()(

)( , and (2.64)

zi abezAidz

zdB )()(

)( (2.65)

where )(zA and )(zB are field amplitudes as a function of propagation in z-

direction in the respective waveguide. a and b are the propagation constant in

each waveguide whereas is a coupling coefficient. In this case, we assume two

guided modes propagating in the same direction (+z direction), thus

40

0a and 0b . Equations (2.64) and (2.65) indicate the inter-relationship of

the respective field amplitudes in each waveguide. With the initial condition

1)0( A and 0)0( B which correspond to the one optical input, the solution to

equations (2.63) and (2.64) can be obtained [4]:

zizezA zi

sincos)( (2.66)

zi

ezB zi

sin)( (2.67)

where ab 2 , corresponding to degree of synchronism between the mode

a and b. The parameter defined as 2/122 . From the solution above,

it can be seen that the propagation mode (energy) will transmit back and forth

between two waveguides in a periodic manner. The complex number indicates

the phase change occurring each time the mode is transferred to another

waveguide. It is also apparent seen that the coupling coefficient plays an

important role in coupling activities.

The power flow in waveguide described by 2

)(zA and 2

)(zB and can be

expressed as:

zFA

zA2

2

2

sin1)0(

)( (2.68)

zFA

zB2

2

2

sin)0(

)( (2.69)

41

where 2

2

)/(1

1

F .The minimum distance to achieve a complete

coupling process (maximum power transfer) is defined as the coupling length,

cL given by [4]:

2

mLc , for integer values of m (2.70)

where is the coupling coefficient of the two waveguides. The coupling

coefficient is a parameter used to measure the degree of overlap that occurs

between the evanescent fields of each waveguide [4]. Based on mode

interference phenomena, the coupling coefficient can be acquired by

analyzing both even and odd modes in the waveguides. Therefore, according to

the mode interference phenomena, the minimum distance required for the

complete transfer of light energy from one input waveguide to another

waveguide is:

oe

cL

(2.71)

where e and o are the propagation constant for even and odd modes,

respectively. Hence the coupling coefficient obtained from equation (2.69) and

(2.70) is given by [4]:

42

2

oe

(2.72)

Therefore, the CMT is a crucial and important theory for designing and

optimizing passive optical planar waveguide device in this work.

2.6 Summary

In this chapter, waveguide analysis methods were described. Initially, the chapter gave a

brief description of planar waveguide structure. The buried waveguide is utilized

throughout the work. The light behavior was elaborated in terms of ray optics and

electromagnetic theory. The ray optics approach only can be utilized for qualitative

description. Therefore, electromagnetic theory is applied to explain the relationship

between electric and magnetic field distribution. Some derivations work on obtaining

EM field distribution has been carried out. However the theoretical approaches above is

only used for slab waveguide and there is necessary to use numerical method solver

such as Finite Difference Method to solve the wave equation for buried waveguide.

Finite Difference Beam Propagation Method (FDBPM) was employed to investigate

light propagation in the buried waveguide. Transparent Boundary Condition (TBC) was

imposed to FDBPM to reduce reflection from incoming light. Finally, the Coupled

Mode Theory (CMT) was introduced and discussed. Coupling length, cL give a rough

indication for designing and optimizing the pump/signal multiplexer in the next chapter.

43

2.7 References

[1] Keigo Iizuka, “Element of Photonics Volume II,” John Wiley & Son Inc., 2002.

[2] H. Kogeinik, “Theory of Optical Waveguides,”Guided-Wave Optoelectronics,

1990.

[3] A. Ghatak and K. Thyagarajan, “Introduction to Fiber Optics, 2nd

ed,”

Cambridge Press, USA, 2000.

[4] K. Okamoto, “Fundamentals of optical waveguide,” Academic Press, USA,

2006.

[5] Gin´es Lifante, “Integrated Photonics: Fundamentals,” John Wiley & Son Ltd,

2002.

[6] G. B. Hocker and W. K. Burns, “Mode dispersion in diffused channel

waveguides by the effective index method,” Applied Optics 16, 113-118 (1977).

[7] Qian Wang, Gerald Farrell, Thomas Freir,” Effective index method for planar

lightwave circuits containing directional couplers, Optics Communications,”

Optics Communications 259(1), 133-136 (2006).

[8] E. A. J. Marcatili, “Dielectric rectangular waveguide and directional coupler for

integrated optics,” Bell Systems Tech. Journal 48, 2071-2102 (1969).

[9] MS Stern, “Finite difference analysis of planar optical waveguides,” Methods for

modeling and simulation of guided-wave optoelectronic devices: modes and

couplings PIER 10, EMW Publishing (1995) .

[10] R. Scarmozzino, A. Gopinath, R. Pregla, S. Helfert, “Numerical Techniques for

Modeling Guide-Wave Photonics Devices,” IEEE Journal of Selected Topics in

Quantum Electronics 6(1), 150-162 2000.

44

[11] J. Jin, “The Finite Element Method in Electromagnetics, 2nd ed.,” New York:

John Wiley & Sons, 2002.

[12] J. S. Gu, P.A. Besse, and H. Melchior, “Method of lines for the analysis of the

propagation characteristics of curved optical rib waveguides,” IEEE J. Quantum

Electron. 27(3), 531-537 1991.

[13] Kenji Kawano and Tsutomu Kitoh, “Introduction to optical waveguide analysis:

Solving Maxwell’s equation and the Schrodinger equation,” John Wiley & Son,

2001.

[14] H. J. W. M. Hoekstra, “Theory and Numerical Strategies of BPMs: On beam

propagation methods for modelling,” Optical and Quantum Electronics 29, 157-

171 1997.

[15] F. Fogli, G. Bellanca, P. Bassi, “TBC and PML conditions for 2D and 3D BPM:

a comparison,” Optical and Quantum Electronics 30, 443-456 (1998)

[16] P. Pantelakis, and E. E. Kriezis, “Modified two-dimensional fast Fourier

transform beam propagation method for media with random variations of

refractive index,” J.Opt.Soc.Am.A, 13(9), 1884-1890 (1996).

[17] B. A. M. Rahman and J. B. Davies, “Finite element analysis of optical and

microwave problems,” IEEE Trans. Microwave Theory Tech 32, 20-28 (1983).

[18] J. Yamauchi, J. Shibayama, O. Saito, O. Uchiyama, and H. Nakano, “Improved

Finite-Difference Beam-Propagation Method Based on the Generalized Douglas

Scheme and Its Application to Semivectorial Analysis,” J.Lightwave Technol.

14(10), 2401-2406 (1996).

[19] Trevor M. Benson, E. V. Bekker, Ana Vukovic, and Phillip Sewell, “Challenges

for integrated optics design and simulation,” Proc. SPIE 6796, 67963C (2007).

45

[20] Robert Scarmozzino, “Simulation tools for devices, systems, and networks,”

Optical Fiber Telecommunications V B: Systems and Networks, 803-863, 2008.

[21] S.T. Chu, W.P. Huang and S.K. Chaudhuri, “Simulation and analysis of

waveguide based optical integrated circuits,” Computer Physics

Communications 68, 451-484 (1991).

[22] G.L. Yip, “Design Methodology for Guided-Wave Photonic Devices,” The

handbook of Photonics, 2007.

[23] Rsoft’s BeamPROP Version 6.0 manual.

[24] Graham T. Reed, and Andrew P. Knights, “Silicon Photonics: An Introduction,”

John Wiley & Sons, 2004.

[25] Tamir Theodor, “Guided-Wave Optoelectronics,” Springer-Verlag, 1990.

[26] Han, X.y., F.f. Pang, et al., "Characteristics of 980/1550 nm WDM coupler

based on planar curved waveguides," Optik-International Journal for Light

and Electron Optics 119(2), 69-73 (2006).